$\int {\mathrm{e}}^{\mathrm{x}} . {\mathrm{x}}^5\mathrm{dx}$ is

• e^x[x⁵ + 5x⁴ + 20x³ + 60x² + 120x + 120] + C

• e^x[x⁵ - 5x⁴ - 20x³ - 60x² - 120x - 120] + C

• e^x[x⁵ - 5x⁴ + 20x³ - 60x² + 120x - 120] + C

• e^x[x⁵ + 5x⁴ + 20x³ - 60x² - 120x + 120] + C

A unit vector perpendicular to both the vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{j}} + \hat{\mathrm{k}}$

• $\frac{- \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{ \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{3}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

If $\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = \overrightarrow{\mathrm{a}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \overrightarrow{\mathrm{a}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 1, then  $\overrightarrow{\mathrm{a}}$ is equal to

• $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

• $\hat{\mathrm{i}} - \hat{\mathrm{k}}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are unit vectors and $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = 1$, then $\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|$ is equal to

• $\sqrt{2}$

• 1

• $\sqrt{5}$

• $\sqrt{3}$

The projection of $\overrightarrow{\mathrm{a}} = 3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{on} \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ is

• $\frac{8}{\sqrt{35}}$

• $\frac{8}{\sqrt{39}}$

• $\frac{8}{\sqrt{14}}$

• $\sqrt{14}$

The value of ${\int }_{- 2}^2\left({\mathrm{ax}}^3 + \mathrm{bx} + \mathrm{c}\right)\mathrm{dx}$ depends on the

• value of b

• value of c

• value of a

• values of a and b

The area of the region bounded by y = 2x - x² and the x-axis is

• $\frac{8}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{4}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{7}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{2}{3} \mathrm{sq} \mathrm{unit}$

18.

The differential equation $\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{x} = \mathrm{c}$ represents

• a family of hyperbolas

• a family of circles whose centres are on the y-axis

• a family of parabolas

• a family of circles whose centres are on the x-axis

$\int \frac{\sec \left(\mathrm{x}\right)}{\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\tan \left(\mathrm{x}\right) - \sec \left(\mathrm{x}\right) + \mathrm{C}$

• $\log \left(1 + \sec \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $\log \left(\sin \left(\mathrm{x}\right)\right) - \log \left(\cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right), \mathrm{then} \int \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{2}{\mathrm{f}}^2\left(\mathrm{x}\right)$

• $\frac{1}{2}{\mathrm{g}}^2\left(\mathrm{x}\right)$

• $\frac{1}{2}{\left[\mathrm{g}\text{'}\left(\mathrm{x}\right)\right]}^2$

• f'(x)g(x)